New Results for the 2-Interval Pattern Problem

We present new results concerning the problem of nding a constrained pattern in a set of 2-intervals. Given a set of n 2-intervals D and a model R describing if two disjoint 2-intervals can be in precedence order (<), be allowed to nest (@) and/or be allowed to cross (G), the problem asks to nd a maximum cardinality subset D′ ⊆ D such that any two 2-intervals in D′ agree with R. We improve the time complexity of the best known algorithm for R = {@} by giving an optimal O(n log n) time algorithm. Also, we give a graph-like relaxation for R = {@, G} that is solvable in O(n2√n) time. Finally, we prove that the problem is NP-complete for R = {<, G}, and in addition to that, we give a xedparameter tractability result based on the crossing structure of D.